The Plateau problem, named by Henry Lebesgue after the Belgian physicist, consists in finding the surface of least area which spans a given contour. In order to tackle such question, generations of mathematicians have investigated the very fundamental notions of ``surface'', ``boundary'' and ``area'', proposing a variety of different theories. In this talk I will give a brief exposition of the so-called theory of currents, introduced by Federer and Fleming in the 60es after the pioneering work of De Giorgi in the case of hypersurfaces. I will then discuss an open question relating the shapes of the contour and that of the minimizer, posed by Almgren in the early eighties and recently solved in a joint work with Guido de Philippis, Jonas Hirsch and Annalisa Massaccesi.